MATH6030 (Spring 2026): Introductory topics in Representation theory.

Instructor: Prof. Ivan Loseu (email: ivan.loseu@yale.edu)

Lectures: MW 1-2.15, Location: TBA. The first class is on 1/21 and the second is on 1/23. The last class will be held during the reading week.

Office hours: M 3.30-4.30, TF 10.30-11.30 in KT 715.

The class discusses several topics in Representation theory including those of current interest. This document contains important information on the class content, prerequisites, homeworks, references, etc. Please read it! More references will be posted in the document as the class progresses. In particular, all references below are to the

A brief (and very condensed) write-up on the basics of Representation theory. For a more detailed treatment, please see the references for the course.

A "map" of this class.

Homeworks:

  • Homework 1, due Wednesday, Feb 18.
  • Homework 2, due Monday, Mar 30.

    Schedule:

  • Jan 21, Lecture 1: Algebraic groups and Lie algebras I, notes. We discuss some basics of Algebraic geometry and then define algebraic groups and their rational representations and give examples. References include [OV], 2.1.1-2.1.4, 3.1.1; [H1], 1.1-1.5, 7.1, 7.2, 7.4.
  • Jan 23, Lecture 2: Algebraic groups and Lie algebras II, notes. We discuss tangent spaces in Algebraic geometry, compute the tangent spaces at 1 for the classical groups and then state a result about structures on the tangent spaces to algebraic groups at 1. References include: [H1], Sections 5,9. [OV], Sec. 1.2 (in the context of Lie groups).
  • Jan 26, Lecture 3: Algebraic groups and Lie algebras III, "official" notes as well notes written during the zoom lecture. We introduce distribution algebras of algebraic groups and use them to prove the main theorem in Lecture 2.
  • Jan 28, Lecture 4: Algebraic groups and Lie algebras IV, notes: we finish the proof of the theorem from Lecture 2. Our main topic is the Lie algebras, their representations and the universal enveloping algebra. References: [H2], Sections 1,17; [H1], Section 10.
  • Feb 2, Lecture 5: Algebraic groups and Lie algebras V/ SL_2 and sl_2, I, notes: we finish our discussion of the universal enveloping algebra and then talk about connections between representations of algebraic groups and of their Lie algebras. After this we define simple algebraic groups and Lie algebras, that are most interesting from the point of view of Representation theory. And we start discussing the representation theory of sl_2 in characteristic 0. References: [H2], Section 17, [H1], Section 13.
  • Feb 4, Lecture 6: SL_2 and sl_2, II, notes: In the case of characteristic 0, we classify the finite dimensional representations of sl_2 and prove (time permitting) that all finite dimensional representations are completely reducible. References: [H2], Secs 6,7.
  • Feb 9, Lecture 7: SL_2 and sl_2, III, notes: We discuss the representation theory of sl_2 in positive characteristic. Reference: the original paper by Rudakov and Shafarevich.
  • Feb 11, Lecture 8: SL_2 and sl_2, IV, notes: We finish the classification of irreducible representations of sl_2 and start talking about rational representations of SL_2. References: the Rudakov-Shafarevich paper for the first section. This text is a expanded version of the notes of the previous version of the course.
  • Feb 16, Lecture 9: SL_2 and sl_2, V, notes: We discuss the "highest weight theory" for rational representations of SL_2 and start proving the classification of irreducible representations. References: the same notes as for the previous lecture. Also see Sec. 5.8 in [E] for induction of representations.
  • Feb 18, Lecture 10: Hopf algebras, filtrations and gradings, I, notes: We finish the classification of irreducible representations of SL_2 and start a new topic: we'll define Hopf algebras and give examples. A reference for Hopf algebras is 4.1, A and B, in [CP].
  • Feb 23, Lecture 11: Hopf algebras, filtrations and gradings, II, "official" notes, and in-class notes: We discuss graded and filtered vector spaces explaining basic constructions and examples. Then we start applying this formalism to proving the PBW theorem.
  • Feb 25, Lecture 12: Hopf algebras, filtrations and gradings, III, notes: We use Hopf algebras, filtrations and gradings to establish Facts B (PBW theorem), and D listed in Lecture 10. Then we start discussing comodules over coalgebras.
  • Mar 2, Lecture 13: Hopf algebras, filtrations and gradings, IV: we continue our discussion of comodules. We explain how a rational representation is a module over the distribution algebra and discuss a connection between the distribution algebra and the universal enveloping algebra. Finally, we explain how to construct a faithful representation of an algebraic group.